Product type bounds on the approximation of values of E- and G-functions
This paper obtains effective bounds on the absolute values of products of linear forms, with integral coefficients, in values of certain Siegel E- or G-functions. The results are given in terms of the product of the absolute values of the coefficients, and differ somewhat from those obtained recently by \textit{G. V. Chudnovsky} [Proc. Natl. Acad. Sci. USA 80, 3138-3141 (1983; Zbl 0511.10027)], \textit{G. V.} and \textit{D. V. Chudnovsky} [ibid. 80, 5158-5162 (1983; Zbl 0516.12019); Adv. Math. 53, 28-54 (1984; Zbl 0539.10029)]. Consider the case of one form, which is the most satisfactorily understood case, and suppose that we are considering values of Siegel E- functions. G. V. and D. V. Chudnovsky obtained bounds that depended upon the maximum of the absolute values of the nonzero coefficients raised to the power 1-(the number of values being considered)-epsilon, for each epsilon greater than zero, and not on the product of the nonzero coefficients. The present paper obtains bounds that depend upon the product of the absolute values of the nonzero coefficients raised to the power 1-(the number of values appearing in the form) - epsilon, for each epsilon greater than zero. The bounds prior to the work of the Chudnovskys' and the present paper depended upon the maximum of the values of a set of coefficients that included possibly zero coefficients. (This did not affect the maximum, of course, but unfortunately the exponent then had to equal 1-(the cardinality of this possibly larger set of coefficients)- epsilon, for each epsilon greater than zero.
scientific article; zbMATH DE number 3952874
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